Beyond Magnitude: Leveraging Direction of RLVR Updates for LLM Reasoning

Abstract

Reinforcement learning with verifiable rewards (RLVR) has substantially improved the reasoning capabilities of large language models. While existing analyses identify that RLVR-induced changes are sparse, they primarily focus on the **magnitude** of these updates, largely overlooking their **direction**. In this work, we argue that the direction of updates is a more critical lens for understanding RLVR's effects, which can be captured by the signed, token-level log probability difference $\Delta\log p$ between the base and final RLVR models. Through statistical analysis and token-replacement interventions, we demonstrate that $\Delta\log p$ more effectively identifies sparse, yet reasoning-critical updates than magnitude-based metrics (e.g., divergence or entropy). Building on this insight, we propose two practical applications: (1) a *test-time extrapolation* method that amplifies the policy along the learned $\Delta\log p$ direction to improve reasoning accuracy without further training; (2) a *training-time reweighting* method that focuses learning on low-probability (corresponding to higher $\Delta\log p$) tokens, which improves reasoning performance across models and benchmarks. Our work establishes the direction of change as a key principle for analyzing and improving RLVR.